Monday, November 26, 2012

North Celestial Pole 03b

  I again used the rotation center formula to calculate the pixel position of the center of rotation in the image using the pairs of points within 1000 pixels of the approximate center. The root mear square error for the pair distances from the center was 1.59 pixels. The RA and Decl are known for α UMi and λ UMi and they can be used to obtain a scale factor which can be used to convert pixel separations to angular distances.
  Letting e be a unit vector representing the position of a star and using the formula equating the dot product of two positions to the cosine of the angle between them we get two conditions for direction of the pole. Each condition specifies a circle on the celestial sphere with the known angular distance of the pole from the star. We can represent the circle about λ UMi by an equation for a circle on the sphere.
  The second condition tells us that the position of the pole is a zero of a nonlinear equation which can be plotted to estimate the angles for the zeros.
  Substituting the equation for the circle from the first condition into the second condition we get a simple equation which is linear in terms of the sine and cosine of an angle that can be solved for an expression for the sine of the unknown angle.
  Plugging the constants into the expression gives the sine of the angle, the angle and the direction of the pole in Cartesian coordinates.
  We can now calculate the Right Ascension and Declination of the computed pole in the J2000 coordinate system.
  Twelve years have elapsed since the year 2000 and we can expect some movement of the pole due to precession and nutation. The observed pole is about 9 arc minutes from the J2000 pole. The expected error in the observed pole is about 14 arc seconds.

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